Saturday, September 04, 2010 ... /////

Why Nature loves exact symmetries

Symmetries and beauty have been analyzed many times on this blog. However, in this text, I would like to focus on the question why Nature wants symmetries to be exact.

So why does Nature love to obey exact symmetries? The short answer is "Why not!?". That's a little bit too imprecise a slogan which deserves a clarification:

A symmetric explanation is qualitatively different from an asymmetric explanation. It therefore deserves a comparable prior probability.

Because a symmetric theory is more constraining, an agreement with observations adds more units of evidence in favor of its validity than it does to the competing asymmetric theory.
In other words, if we have two hypotheses that describe the same phenomena, S (which is symmetric) and A (which is its asymmetric generalization), and they agree with the observations equally well, it was a priori more nontrivial for S to agree with the observations.

In other words, when the tests work, the Bayesian formula doesn't reduce the probability that the symmetric theory is valid while the asymmetric theory gets punished because the probability decreases by the factor of the probability that the parameters are fine-tuned to "hide" the apparent symmetry within the experimental error margin.

That sounds like a much more rational explanation of a symmetry than vague reference to beauty and symmetry (that keeps on evolving), doesn't it? But it's still the same thing, expressed in a more mathematical language.

Of course, one must be careful: if there exists some actual strong piece of evidence - direct or indirect - that the more constraining and more symmetric hypothesis, S, is invalid, then S may be falsified. No compassion and no cult of beauty may help if the evidence against S is strong and error-free. For example, beta-decay experiments clearly show that the world is not left-right symmetric.

Let us look at the Bayes formula a little bit more closely. A priori, the two competing hypotheses, S and A, are given comparable priors, say 50% each.

If you're an unbiased scientist, you should always start with prior probabilities that are comparable for "qualitatively different" explanations. In other words, you shouldn't "positively discriminate" one explanation (and suppress others) because it predicts a bigger value of quantity Q, which is your favorite one (e.g. the number of good prayers the belief will bring you in the future, or the number of microstates of the initial state), unless you can actually show that Q is linked to the probability even when your favorite assumption is invalid.

Every qualitatively different opinion or hypothesis has to get a chance - and the evidence is always a sufficiently strong tool to eliminate all the wrong answers, leaving the correct one. But had you excluded the right answer at the very beginning, no one would be able to resurrect it. So:
P(S) = P(A) = 1/2
Observations show that the empirical statement E holds. We have to refine the prior probabilities, P(S) and P(A), to the posterior probabilities which are the conditional probabilities that S or A hold given E.
P(S|E) = P(S) P(E|S)/P(E)
P(A|E) = P(A) P(E|A)/P(E)
If the hypothesis S - that is free of symmetry-breaking parameters - predicts E to (probably) happen, then the P(E|S)/P(E) ratio is comparable to one. On the other hand, the asymmetric hypothesis A has to be divided to many partial hypotheses A_{1}...A_{n}. Only a fraction of them has sufficiently small values of the symmetry-breaking parameters to be compatible with the apparently symmetric observation E. That's why P(E|A) is pretty small - controlled by this fraction.

Let's assume that S and A are complementary: we know that one of them is valid. And let's study what happens with the ratio of their probabilities. Dividing the last two displayed equations, we see that
P(S|E) / P(A|E) = P(S) / P(A) * P(E|S) / P(E|A)

ln[P(S|E)/P(A|E)] = ...
... = ln[P(S)/P(A)] + ln P(E|S) - ln P(E|A)

AEafter E = AEbefore E + ln P(E|S) - ln P(E|A)
For your convenience, I have also included the logarithm of the ratio and reinterpreted this logarithmic equation in terms of the "amount of evidence", AE, for saying that S is true. This amount of evidence has shifted by the two logarithms. In particular, -ln P(E|A) may be pretty large and positive if the asymmetric hypothesis A needs a lot of fine-tuning to agree with the apparently symmetric observations.

Recall that if the "amount of evidence" gets to +14 or so, you will have gained the equivalent of a 5-sigma evidence supporting the hypothesis S.

The logic above may be applied to any kind of fine-tuning, not necessarily one that is related to a symmetry. For example, if you get convinced that the natural a priori distribution for the squared Higgs mass in the pure Standard Model is the uniform distribution on the interval (0,M_{Planck}^2), then P(E|A) for "A" being the Standard Model and "E" being the observed 100-GeV-ish Higgs is something like 10^{-32} (the squared mass ratio: the probability is simply the ratio of the lengths of two intervals on the squared mass scale).

In this case, "-ln P(E|A)" is equal to +74 or so which is more than a sufficient "amount of evidence" to exclude a Standard Model in favor of an alternative theory that predicts a higher probability that the Higgs mass is low relatively to the Planck scale. For example, SUSY is likely to link the Higgs mass to the mass of the superpartners and the superpartner scale may be "naturally" distributed by a uniform distribution on the log energy scale. Instead of ln 10^{32} = 74, you may get ln ln 10^{32} = 4.3 or so which is much smaller an amount of evidence against SUSY.

In the imagined SUSY vs SM battle, SUSY would gain approximately 70 units of evidence because of the hierarchy problem. Note that both probabilities P(E|SM) and P(E|SUSY) - where "E" is a light Higgs observation - are actually proportional to the error margin of the Higgs mass (approximately, if the error margin is sufficiently narrow), so this factor nicely cancels in the ratio. If we know that the Higgs is light, and we kind of do, we can calculate the relative odds of two theories' being right and the result can't depend and doesn't depend on the unknown uncertainty of the Higgs mass.

By all the formulae above, I wanted to convince you that there exists a pretty "canonical" quantitative way to decide which of two competing, qualitatively different theories - simpler vs more general one; more symmetric one vs less symmetric one - is favored by the observations and by how much. If the observations agree with a symmetry, the strictly symmetric theory gets a boost.

Approximately symmetric theories

If a symmetry is observed, a theory A that needs to fine-tune some parameters with a big enough precision would seem to be hugely punished. However, there typically exist "intermediate" theories - which we may view as a refined subclass of A - that claim that the world is fundamentally asymmetric but include an extra level of sophistication to explain why the symmetry violations are small.

Note that this level of sophistication must actually contain some functional mechanisms, at least vague ones, not just a pure wishful thinking, for the probability P(A|E) to get larger.

For example, the baryon number conservation - and the associated U(1) symmetry - is pretty unlikely to be exact in the real world. However, the theory that the symmetry is violated would seem to be excluded by a large amount of evidence because one needs to fine-tune various parameters very accurately. Recall that no proton decay has been seen yet. And the probability that all these coefficients are fine-tuned sufficiently close to zero is small.

However, the refined versions of A - with the baryon number violation - actually do explain why the violations are small. Using the fields and the Lagrangian of the Standard Model, one can prove that there exists the so-called "accidental symmetry". It means that all baryon-violating terms must automatically be non-renormalizable terms! Because they're non-renormalizable terms, their coefficients are naturally of the form "1/Lambda^n" where "Lambda" is a high energy scale where new physics comes (possibly GUT scale) and "n" is a positive exponent.

In plain English, the coefficients of non-renormalizable terms are automatically small and the Standard Model as the effective theory (which allows all the non-renormalizable terms with their natural-magnitude coefficients) does explain why we haven't seen any proton decay etc.

Of course, despite the missing evidence of a proton decay, I do believe - for deeper theoretical reasons - that the baryon number U(1) symmetry is not preserved exactly. When you create a black hole out of a star with a large "B", it will eventually Hawking-radiate into radiation whose baryon number "B" is essentially zero.

(A commenter wittily mentions that aside from the black holes, there exists a more accessible source of dynamics that changes "B": the electroweak sphalerons - kind of instantons with negative modes.)

Among the simple "generalized charges" that are different from the well-known electric charge, only "B-L" has a significant chance to be an exact symmetry - a gauge symmetry - in models beyond the Standard Model (namely GUT theories).

Even this symmetry still has to be broken because there seem to be no long-range forces that attract particles according to their "B-L".

The punch line of this section is that asymmetric theories are not "automatically bad"; what matters is whether there is an explanation - or a chance that it will be found - that clarifies why the symmetry-breaking terms are so small to agree with the apparently symmetric observations. In some cases, there actually exist such explanations (the accidental baryon number symmetry of the Standard Model); in others, it doesn't.

Exact gauge symmetries are fundamental for consistency

Finally, I want to repeat some points that have probably been explained many times on this blog. The point I want to make is that the gauge symmetries have to be exact, otherwise we would face huge troubles with the consistency of our theories. I mean the Yang-Mills symmetries of the Standard Model and the diffeomorphism symmetry of general relativity, among their diverse generalizations and extensions found in string theory.

First, it's important to note that the local Lorentz symmetry is extremely likely to be exact. Why? Because of the arguments explained above: if Nature is fundamentally Lorentz-asymmetric, such a hypothesis would predict many large coefficients of the Lorentz-breaking terms. However, in order to agree with the empirical tests that have seen no Lorentz violation, all these coefficients have to be fine-tuned to zero with a large accuracy.

The probability that it naturally happens is extremely tiny. In fact, even the anthropic principle is unhelpful for this goal because life doesn't seem to depend on the exact Lorentz symmetry, at least not in an obvious way.

Even the "more refined" versions of the asymmetric theories fail. You could conjecture that the Lorentz-breaking operators must be non-renormalizable. But your conjecture would actually be invalid. Coleman and Glashow have found exactly 46 renormalizable CP-even but Lorentz-breaking terms to deform the Standard Model.

Even if the statement about the non-renormalizability is wrong, you could conjecture that the Lorentz-breaking terms are naturally suppressed by "1/Lambda^n" coefficients. However, this conjecture also seems to be invalid. The maximum "Lambda" that can naturally occur in similar terms is the Planck energy. However, the Fermi satellite has excluded even the theories where the coefficients as as small as "1/E_{Planck}".

So the Lorentz symmetry is almost certainly exact.

But there is another interesting interplay between the Lorentz symmetry on one side and the gauge symmetries - and the diffeomorphism group - on the other side. As you know, the Standard Model as well as general relativity respect the exact local Lorentz symmetry. They also have additional, local symmetries of their own.

There are many ways to describe the gauge theories of the Standard Model as well as general relativity: you may gauge-fix their local symmetries, or partially gauge-fix them, or use an S-dual or holographically dual description (without the original gauge symmetries) to obtain the physically equivalent predictions. Because there exist these descriptions, are the Yang-Mills transformations and coordinate redefinitions "real"?

Well, they're not "real" in the physical sense - because the pure gauge degrees of freedom can't be observed, not even in principle. But these symmetries are still damn useful. Why?

Because the usual description with the gauge symmetries is necessary if you want to keep the local Lorentz symmetry of the theory manifest! You start with spin-one gauge fields or spin-two metric tensor field whose covariant behavior under the Lorentz transformations is standard. You may write some Lagrangians for these fields - and it's trivial to construct Lorentz-invariant actions by a proper contraction of the indices.

However, these fields with spin greater than or equal to one inevitably have "time-like" components. The corresponding creation operators - their Fourier transforms - would create states of a negative norm. They would lead to negative probabilities which would be bad for the logical consistency (even if they were just a "little bit negative": but the more negative they can be, the worse). So there must exist a method to get rid of these unphysical degrees of freedom.

A gauge symmetry is needed for every time-like components. You may postulate that the states have to be exactly invariant under the gauge symmetry - which is enough to ban the time-like modes (bad ghosts). In the previous sentence, you need an exact symmetry (that commutes with the Hamiltonian); otherwise, your ban would depend on the moment when you impose it. So the SU(3) gauge field has eight "bicolors" of the A_0 component - which is exactly the same as the number of different parameters of gauge transformations that effectively remove these A_0 fields (together with the longitudinal modes). Recall that the dimension of the SU(3) algebra equals eight.

For general relativity, it's the mixed (spatial-temporal) components "g_{0i}" of the metric tensor that would create negative-norm states ("g_{00}" would bring two minuses i.e. a plus); here, I am imagining that we are expanding around a Minkowskian background which is always "locally" possible to imagine. There is a kind of a vector worth of them - 3 or 4 components per point (let's not discuss the subtle difference between 3 and 4 here) - and they match the number of parameters (four) that identify a diffeomorphism: x1,x2,x3,x4 are four functions of the old coordinates x1',x2',x3',x4'.

So the exact gauge symmetries are necessary to get rid of the unphysical components of the spinning fields - with spin 1, 3/2, or 2 (or higher). However, this argument doesn't mean that these symmetries have to stay unbroken. Quite on the contrary, a spontaneously broken symmetry may eliminate all the problems as well. It will make the gauge bosons massive but the time-like, negative-norm component will still be eliminated. (At very high energies, the effects of any spontaneous breaking go to zero.)

As I have explained in the past, a spontaneously broken symmetry is "as pretty" as an unbroken one. Their relationship is similar to the relationship of the functions
x4 + 2x2   ↔   x4 - 2x2.
They look pretty similar, don't they? But the first function has a minimum at "x=0" (a symmetry-preserving potential) while the second function has two minima at "x=-1" and "x=+1" (a symmetry-breaking potential). You see that mathematically, there's nothing "uglier" about the second function: it just has different physical consequences.

Summary

To summarize, it's legitimate to ask whether a particular symmetry or principle may be broken or violated. However, one should never assume that the answer to such questions will remain "Yes" forever. Quite on the contrary, a considerable amount of evidence may often be collected to show that the answer is almost certainly "No". Nature likes many of Her symmetries to be exact and we are given many opportunities to learn about this secret of Hers.

And that's the memo.